Is $\frac{z-\alpha}{1-\overline{\alpha}z}$ some special function in complex analysis?

Question

Many homework problems seem to use the following function (or something very close to it):

$$F_\alpha(z)=\frac{z-\alpha}{1-\overline{\alpha}z}$$

Does it serve some special purposes in complex analysis?

It does revolve around the unit circle.

it is a special case of a moebius transform – tired Jan 18 '16 at 18:06 — tired, Jan 18 '16 at 18:06

score 3 · Answer 1 · 2016-01-18T18:17:07.173

3

This is "the" conformal map from $\mathbb{D}$ to itself that sends $a$ with $0$, and is unique up to rotation. Applications include things such as the Schwarz-Pick theorem.

edited Jan 18 '16 at 18:17

answered Jan 18 '16 at 18:13

But $F_\alpha(0) = -\alpha$...? – Unit Jan 18 '16 at 18:16
@Unit Sorry, had it worded wrong. – Jan 18 '16 at 18:17

Is $\frac{z-\alpha}{1-\overline{\alpha}z}$ some special function in complex analysis?

1 Answers1